Remember when it was fashionable for rock bands to do an album with a symphony orchestra eg. Kiss, Metallica, Dream Theater etc.
Well those orchestras had clarinets, flutes, violins and many more instruments.
Anyway if you put in lots of orchestral instruments you get chamber pop like Sufjan Stevens or Jeremy Enigk's solo stuff.
I think the softer an instrument sounds, the less likely it is to be used in a rock context. This is why you hear sax but not clarinet. However, I find trumets to be even sharper sounding than sax, and wonder why it hasn't been used in rock like sax has. Then there is flute, which is very soft sounding but still finds its way in here and there.
Bass clarinet has a very synth-like legato to it. I love hearing Eric Dolphy wail on one. I think with a little treatment, the Bass Clarinet can make a stunning soloing device in rock music. As previously mentioned, Zappa certainly liked it.